CANT , and it's effect ?

[nervoustrig]

...take your favorite rifle and do the experiment I discussed earlier, Shoot with your standard scope mounting setup currently in place at some multiple of your near zero distance to determine the necessary holdover for that distance and that scope. Then shoot canted a specific amount using that holdover point marked vertically above the intended POI with the crosshair holdover method aimed at that marked point. Observe and mark the horizontal error seen. Then, mount a noticeably higher scope and repeat the EXACT same process to produce a DIRECT low to high scope comparison of cant error produced by each.

Again, this is a suggestion we can both agree on, because by establishing a proper holdover for each scope and using it for aiming, it will prove there to be no difference in cant error.


The proper holdover can be either a mildot or an equivalent spot on the vertical bar of a simple crosshair. Repeatability with the latter will be a challenge; a mildot reticle will give more confident results.

 

[Scotchmo]

...

... Shoot with your standard scope mounting setup currently in place at some multiple of your near zero distance to determine the necessary holdover for that distance and that scope. Then shoot canted a specific amount using that holdover point marked vertically above the intended POI with the crosshair holdover method aimed at that marked point. Observe and mark the horizontal error seen. Then, mount a noticeably higher scope and repeat the EXACT same process to produce a DIRECT low to high scope comparison of cant error produced by each. The more difference in scope height, the better to see the difference. And the larger the multiple of the near zero distance, also the better to see the difference. ...

"...holdover point marked vertically above the intended POI..."

That is actually a new instance where the aim point has been measured and moved to a new position. It requires me to take a test shot and then to physically go out to the target and measure and mark a new calibrated aim point above the intended POI. Who does that? Even with that reference point, assume I still managed to cant the gun (on purpose). If I cant to the left, the reticle cant error deviates to the right of the vertical reticle and gun cant error deviates to the left of the vertical reticle. Scope cant error now partially (or even wholly) cancels the gun cant error. Voila - the low scope will almost always be better.

By drawing a new aim point on the target (hard to do it while hunting/FT/sniping), you have come up with a way to use reticle cant to partially cancel out gun cant.

Congratulations. I agree that it will behave as you stated. It is a unique variation to #2 but where the new aim point has been pre-plotted rather than improvised along the cant angle.

Two three instances where POA does not match intended POI:

#1) Using mil-dot holdover or clicks that are wrong for the target distance. POA does not match intended POI.

#2) Holding cross hair (aiming) on a point above the intended POI in order compensate for drop. ("improvised holdover"?) POA does not match intended POI.

#3) For far targets, go out to the target and mark a new POA directly above the intended POI, by an amount equal to the difference from your close zero. POA does not match intended POI.

 

[Guest]

NOTE ~~ There are significant errors in my calculations in this post and repeats of it. Further down you will find a corrected Spread Sheet which can be downloaded if you (or anyone else) wants to fix it Mo' Better.

https://www.facebook.com/USAMU1956/videos/635879053972109/

https://www.ssusa.org/articles/2018/5/24/effects-of-cant/

Most of the content here is now proved wrong. No point in leaving it here to further confuse. Deleted.

The reason the horizontal error swamps the vertical error in all cases is intuitively obvious when you realize we are dealing with a triangle which always has the long side oriented vertically at any cant angle. In other words, the drop always swamps the other variables. You could also say that the vertical axis in the reticle becomes the horizontal axis when cant reaches 45 degrees. ...

This graph models my Condor shooting .22 JSB Exact Heavies 18.39 grain @873 fps for about 31 fpe.

The table is more precise than the graph. ... Lol unfortunately it to was bad so I deleted it... =)

Because calculus R hard.

Hopefully this puts the magnitude of this problem in perspective for the casual shooter who can mostly ignore the problem. OTHO If you are shooting BR or long range (for your setup) you better remove as much cant from your shooting platform as you are able.

NOTE:

Tabular data Deleted BAD Example of CGP Laser graph, there is bad math in there. Deleted.

If you want to play with this information in your spread sheet you can do all the intermediate ballistic calculations in CGP. Turn on the LASER. Set zero coincides with trajectory, and height is relative to bore with the bore at the height of your optic and then tell CGP you want LASER to coincide with zero. Finally set only one zero.. The LASER Hold Over graph can be used to pull all the information you need to plug into the equations.

HTH

Mike

EDIT: I found two errors in the text and images both call out the wrong zero range. The zero range on this data is 36 meters. One zero. Scotchmo found additional errors in the formulae I used. I corrected that and uploaded the spreadsheet below.

 

[Scotchmo]

...

This graph models my Condor shooting .22 JSB Exact Heavies 18.39 grain @873 fps for about 31 fpe.

The chart is more precise than the graph. I did not choose to compute any finer granularity than 10 meter increments.

The formula for horizontal error is tan(cant_angle) times mid range trajectory height times four. but check my notes elsewhere.

The formula for vertical error is (1/2) tan(cant_angle) times horizontal error. This is a shortcut (approximation) which ignores the fact that a circle is not polygon. Because calculus R hard.

"...horizontal error is tan(cant_angle) times mid range trajectory height times four..."

"...vertical error is (1/2) tan(cant_angle) times horizontal error..."

"...This is a shortcut (approximation) which ignores the fact that a circle is not polygon. Because calculus R hard...."

It is possible to model it as the circle rather than the polygon approximation. Your equations could give us good enough answers when dealing with constant velocity projectiles at relatively small gun cant angles. What does it take to get a precise answer for a full range of gun cant angles?

For a constant velocity projectile: 4x mid_range_drop = full range drop

Fortunately, I can use Chairgun to give me the full range drop (drop from bore line). That factors in the decaying velocity of the high BC pellets. And it means I can avoid dealing with the calculus. We are then left with a fairly simple geometry/trig problem.

You used tan instead of sin, and that is fine, as in this case they are almost the same, but only for relatively small gun cant angles like 5 degrees. sin will give the correct horizontal values for the full range of gun cant angles (0 to 180 degrees).

You used tan instead of cos for the other axis, and that is also fine for small gun cant angles. cos will give the correct vertical values for the full range of gun cant angles.

Replace (4x mid_drop) with drop, and replace tan with sin and cos. Now we have:

horizontal error = drop x sin(cant_angle)

vertical error = drop x (1-cos(cant_angle))

 

[Guest]

... House cleaning my cruft ... It was wrong anyway. =)

"...horizontal error is tan(cant_angle) times mid range trajectory height times four..."

"...vertical error is (1/2) tan(cant_angle) times horizontal error..."

"...This is a shortcut (approximation) which ignores the fact that a circle is not polygon. Because calculus R hard...."

It is possible to model it as the circle rather than the polygon approximation. Your equations could give us good enough answers when dealing with constant velocity projectiles at relatively small gun cant angles. What does it take to get a precise answer for a full range of gun cant angles?

For a constant velocity projectile: 4x mid_range_drop = full range drop

Fortunately, I can use Chairgun to give me the full range drop (drop from bore line). That factors in the decaying velocity of the high BC pellets. And it means I can avoid dealing with the calculus. We are then left with a fairly simple geometry/trig problem.

You used tan instead of sin, and that is fine, as in this case they are almost the same, but only for relatively small gun cant angles like 5 degrees. sin will give the correct horizontal values for the full range of gun cant angles (0 to 180 degrees).

You used tan instead of cos for the other axis, and that is also fine for small gun cant angles. cos will give the correct vertical values for the full range of gun cant angles.

Replace (4x mid_drop) with drop, and replace tan with sin and cos. Now we have:

horizontal error = drop x sin(cant_angle)

vertical error = drop x (1-cos(cant_angle))

Understood

I used tan in the horizontal displacement because I thought the second link called for that. I have reread that and I agree I should have read more closely. It calls for sine times drop.

I used tangent in the vertical displacement because I was using the horizontal error to solve a right triangle across the cord subtended by the cant-angle. Might be wrong, like I said I don't really speak math, I just pretend too. I do think it is a reasonable approximation in all cases because it depends on the horizontal error at the target.

I did use CGP to calculate the real drop ballistically. That is what I was explaining how to do with the LASER in CGP. Just didn't make it clear. That eliminated the need to even bother with 4 x max ordinate but like I said... I just pretend I do math. Someone who understands it better almost always comes along to correct it for me.

One other thing I think worth mentioning. These equations assume there is four times as much drop happening after the max ordinate then there is before it. It assumes the projectile arrives at the max ordinate in time = T but takes takes 2* T+T to arrive at the target. The factor 4 is present because drop is per second squared. This is an approximation of course but a pretty good one. So it is not a "constant velocity" solution as you suggest. It is actually approximating the d(v) by using the limits of the equation to calculate drop, right?

1-(cos(2)) is a really small number 6.09172e-4

Here is your new graph using your formulae:

What do these graphs tell you now?

The most significant error I made was not dividing the total drop by 4 in the calculation of horizontal displacement. I (for whatever reason) used 2 instead and that doubled my errors on both computations. Anyway Clearly both of you understand the problem better than I do. I don't really have a dog in this fight. I just figured a tool might cool the heat a bit. Now you two can get back to business with something which at least can be agreed upon.

Also you are correct, the errors would have been MUCH larger had I calculated for high cant angles such as we saw in the first video I linked; however, at no time does the vertical displacement become greater than the horizontal displacement. And correct me if I am wrong but I believe that is because the vertical recticle bar does indeed become a horizontal reference when it passes 45 degrees. It's a circle. Sine and cosine swap values at 90 degrees? 

Here is a copy of the Spread Sheet it is open office.

View attachment cant.1619912295.ods

 

[Scotchmo]

Cornpone,

Good info but still one problem. It appears that you used the intended POI (trajectory from a 30yd zero) rather than the drop. POI (point of impact) is the distance from the LOS (line-of-sight). Drop is the distance from the bore-line. Your use of POI in those formulas, produces a model that is immune to cant errors at the sight-in distance, so it does not quite match a true model. Gun cant errors always get worse with increasing distance. They never get better. I used your spreadsheet (thanks for posting it) and replaced the Trajectory (POI) values with the Chairgun drop values for your gun. I did not check every cell but I think all your other cell values and formulas are correct. Images shown for corrected spreadsheet and resulting graphs.

The POI shown in ballistics programs is only valid when there is no gun cant. As soon as you cant the gun, the POI changes (actual POI diverges from the intended POI). The drop with respect to the bore-line is constant for any given distances, regardless of the amount of cant.

 

[lbc_PSI]

The problem with the tables, is they reflect cant error using the drop for the shooting system with 0 degree cant or 90 degree cant. 

When you cant the shooting system, the elevation angle of the LOB changes with the degree of cant. As you cant from 0 to 90, the LOB elevation angle goes from "X" to 0. 

You cannot use the same trajectory drop for all LOB angles as you go from 0 to 90 shooting system cant. Each new LOB elevation angle will create a new trajectory form. Each new trajectory form will have a new and different drop value for each distance. Each new trajectory form at any given distance is further from the shooting system LOS.

 

[Guest]

Cornpone,

Good info but still one problem. It appears that you used the intended POI (trajectory from a 30yd zero) rather than the drop. POI (point of impact) is the distance from the LOS (line-of-sight). Drop is the distance from the bore-line. Your use of POI in those formulas, produces a model that is immune to cant errors at the sight-in distance, so it does not quite match a true model. Gun cant errors always get worse with increasing distance. They never get better. I used your spreadsheet (thanks for posting it) and replaced the Trajectory (POI) values with the Chairgun drop values for your gun. I did not check every cell but I think all your other cell values and formulas are correct. Images shown for corrected spreadsheet and resulting graphs.

The POI shown in ballistics programs is only valid when there is no gun cant. As soon as you cant the gun, the POI changes (actual POI diverges from the intended POI). The drop with respect to the bore-line is constant for any given distances, regardless of the amount of cant.

Thanks for fixing that. I was going to do it after I read the first line and then saw you had done it for me. Clearly we do not want our laser to have a zero range.

So at this point we are talking about differences of 1/2 cm at 200 meters vertically and about 5 cm at 200 meters horizontally. Those were corrected by getting better drop data.

The model is improving. I'd say it is good enough for AG purposes.

Let's see according to our new model (thanks for fixing that)...

Wind drift at 200 meters for a 1 mph cross wind with the given projectile data is 17.5 cm. That is 8 times as large as the vertical error and about 40% of the horizontal error. A two mile per hour wind would totally swamp cant errors from a 5 degree cant angle.

 

[Guest]

The problem with the tables, is they reflect cant error using the drop for the shooting system with 0 degree cant or 90 degree cant. 

True.

When you cant the shooting system, the elevation angle of the LOB changes with the degree of cant. As you cant from 0 to 90, the LOB elevation angle goes from "X" to 0. 

True. But the absolute drop does not change. It is always the same no matter what the LOB actually is. The drop never changes. Therefor the magnitude and direction of the errors calculated are correct. What has moved is actually the point of aim. That can be calculated and subtracted out but it will not change the numbers we get when running the calculation as we have run it.


You cannot use the same trajectory drop for all LOB angles as you go from 0 to 90 shooting system cant. Each new LOB elevation angle will create a new trajectory form.

Yes you can. We are not computing a firing solution here. We are computing the magnitude and direction of an error. The absolute drop never changes therefore the trajectory never changes. It TRANSLATES around the circle. That impacts the firing solution NOT the amount of error you have to input after you resolve your new aiming point. What I am trying to say is that the displacement of point of aim is a separate problem from the effects of cant. That displacement is NOT dependent upon drop, TOF, or any variable other than a dx, dy which can be calculated and proven to be dependent ONLY upon range to target. It is a different problem.

Right?

 

[Scotchmo]

The problem with the tables, is they reflect cant error using the drop for the shooting system with 0 degree cant or 90 degree cant. 

When you cant the shooting system, the elevation angle of the LOB changes with the degree of cant. As you cant from 0 to 90, the LOB elevation angle goes from "X" to 0. 

You cannot use the same trajectory drop for all LOB angles as you go from 0 to 90 shooting system cant. Each new LOB elevation angle will create a new trajectory form. Each new trajectory form will have a new and different drop value for each distance. Each new trajectory form at any given distance is further from the shooting system LOS.

A picture might help explain:

There are three vectors of concern when shooting a gun.

T1-T2 = target vector (muzzle to target)

S1-S2 = LOS (line-of-sight)

B1-B2 = bore-line

When you cant a gun, it is rotated/canted about a point (T2, S2) which is also the end of the LOS vector, and the intended POI (T2). That point (T2, S2) does not change when canting the gun.

The drop from the end of the bore-line (point B2) is constant for a given distance, but the position of point B2 is shifted when the gun is canted.

When canting the gun, the vector S1-S2 is of no concern, however, it is important when sighting the gun (aiming). And it will determine how much sight adjustment is needed for various distances.

Note: if using a method of aiming where the POA is different than the intended POI, then point S2 is not coincident with T2. Actual POI will be affected accordingly.

 

[Scotchmo]

... What I am trying to say is that the displacement of point of aim is a separate problem from the effects of cant. That displacement is NOT dependent upon drop, TOF, or any variable other than a dx, dy which can be calculated and proven to be dependent ONLY upon range to target. It is a different problem.

Right?

Right!

"...displacement of point of aim is a separate problem from the effects of cant...."

I call that separate problem "scope cant" and it is indeed separate from what I call "gun cant". And yes, it is possible to have both gun cant and scope cant present at the same time.

 

[Guest]

... What I am trying to say is that the displacement of point of aim is a separate problem from the effects of cant. That displacement is NOT dependent upon drop, TOF, or any variable other than a dx, dy which can be calculated and proven to be dependent ONLY upon range to target. It is a different problem.

Right?

Right!

"...displacement of point of aim is a separate problem from the effects of cant...."

I call that separate problem "scope cant" and it is indeed separate from what I call "gun cant". And yes, it is possible to have both gun cant and scope cant present at the same time.

Yes, there is as mentioned somewhere in this thread, a "tolerance" associated with the placement of the reticle in the tube during manufacture. Today when mounting my Crimson Trace I realized there was a very minor but observable difference between the lines on the side of the optic and the actual orientation of the reticle in the tube. This is your "scope cant", which can also occur by rotating the optic within the rings. Scope cant occurs around the axis of the optic. Gun cant occurs around the axis of the bore.

You could also have Scope cant and not realize it. In that case you might be in the habit of checking the plumb of your reticle before a shot against a tree or some such thereby introducing gun cant without realizing it.

By George I do believe I am close to GROKing the problem.

We could add the correction for scope cant into the spread sheet. It would need to be recalculated for each range but that's pretty easy to do in a spread sheet. It could also be shown as a separate part of the solution for cant. If it were handled separately you could input 0 for the scope cant value and the correct answer for rifle cant should fall out of the equations anyway. Scope cant adds or subtracts from rifle cant but is range (and rifle geometry) dependent. Scope height and distance between shooter's eye and the muzzle should be sufficient rifle geometry to calculate it given range. I had to think about that a moment. The angle is actually formed between the shooter's eye and the muzzle, not the turrets and the muzzle. The optic is INSIDE the problem space.

LAST EDIT: Nope, you need to know the distance between the shooter's eye, the turrets, and the muzzle.

 

[Scotchmo]

Scope cant occurs around the axis of the optic. Gun cant occurs around the axis of the bore.
LAST EDIT: Nope, you need to know the distance between the shooter's eye, the turrets, and the muzzle.

Scope cant occurs around the axis of the optic...." - True. Around vector S1-S2 shown above.

"...Gun cant occurs around the axis of the bore...." - not quite. It's actually around point T2 and S2 in the target plane.

In real life gun configurations at typical distances, the angle between the sight vector S1-S2 and the target vector T1-T2 is a fraction of a degree, so those difference are lost in the noise. The end of the rotation axis is at the S2 and T2 points. And that is the point in the target plane that my cant model uses as the point where the gun is canted about.

BTW: If we were actually able to cant the gun about the bore axis B1-B2, the POA through the scope changes, while the POI stays the same. Projectile still hits the bullseye.

If we rotate about the LOS S1-S2 or target vector T1-T2, the POA stays the same while the POI changes. Now we have a gun cant error and miss the bullseye.

 

[Guest]

Scope cant occurs around the axis of the optic. Gun cant occurs around the axis of the bore.
LAST EDIT: Nope, you need to know the distance between the shooter's eye, the turrets, and the muzzle.

Scope cant occurs around the axis of the optic...." - True. Around vector S1-S2 shown above.

"...Gun cant occurs around the axis of the bore...." - not quite. It's actually around point T2 and S2 in the target plane.

In real life gun configurations at typical distances, the angle between the sight vector S1-S2 and the target vector T1-T2 is a fraction of a degree, so those difference are lost in the noise. The end of the rotation axis is at the S2 and T2 points. And that is the point in the target plane that my cant model uses as the point where the gun is canted about.



BTW: If we were actually able to cant the gun about the bore axis B1-B-2, the POA through the scope changes, while the POI stays the same. Projectile still hits the bullseye.

If we rotate about the LOS S1-S2 or target vector T1-T2, the POA stays the same while the POI changes. Now we have a gun cant error and miss the bullseye.

Yes, I see that now. You are correct. In gun cant the bore actually moves around the POA causing the POI to move. That is clear from your earlier diagram and the calculations for that matter. I just did not see it. It has to be that way or there would be no change in POI.

Ok so now we have what I think is a clear agreement on the parameters of the problem and a partially working spread sheet. Are you going to put the changes for scope cant in the spread sheet or do you want me to do that? We might as well finish the job. The information would be useful to FT shooters once they have their systems all locked down. They do have forced shooting positions in that sport and some of those will no doubt include positions which force cant into the equations.

I'd like to see you update this drawing as well. I am not able to do it justice.

I can't think of anything else that needs to be addressed other than the changes to the spread sheet. Do you have more?

Good job on the drawing below. Very clear.

 

[Scotchmo]

...

... Are you going to put the changes for scope cant in the spread sheet or do you want me to do that? ...

...I'd like to see you update this drawing as well. I am not able to do it justice....

I think the spreadsheet demonstrates what we wanted as-is. Scope cant (i.e. rotating scope in the rings) is really a separate issue with a different fix. It should be in a separate spreadsheet in my opinion, or combined in a multidimensional spreadsheet if you want to get fancy.

To show everything clearly in a 2D drawing doesn't work well. Your additions added some more clarity without over-complicating it. I did a 3D model but even that does not really make it clear in a single view. It needs a 3d animation that shows the tracking of the POI as we swing (cant) around the POA. At this point, I don't need an animation in order to understand the problem, but it would be a good educational tool for others.

 

[Guest]

...

... Are you going to put the changes for scope cant in the spread sheet or do you want me to do that? ...

...I'd like to see you update this drawing as well. I am not able to do it justice....

I think the spreadsheet demonstrates what we wanted as-is. Scope cant (i.e. rotating scope in the rings) is really a separate issue with a different fix. It should be in a separate spreadsheet in my opinion, or combined in a multidimensional spreadsheet if you want to get fancy.

To show everything clearly in a 2D drawing doesn't work well. I did a 3D model but even that does not really make it clear in a single view. It needs a 3d animation that shows the tracking of the POI as we swing (cant) around the POA. At this point, I don't need an animation in order to understand the problem, but it would be a good educational tool for others.

There is only so much you can do sometimes.

Well, I'll have a look at fixing that spread sheet maybe this evening if shooting doesn't get in the way. I don't see any other parts of the problem which we can hash out, do you?

 

[Scotchmo]

I don't see any other parts of the problem which we can hash out, do you?

There is that minor omission of the included angle between the LOS and the target vector to muzzle (angle S1-S2-T1). But as I said earlier, it gets lost in the noise.... tan(0.1 degree) vs sin(0.1 degree)

So nothing more to hash out on the gun cant issue.

 

[Guest]

I don't see any other parts of the problem which we can hash out, do you?

There is that minor omission of the included angle between the LOS and the target vector to muzzle (angle S1-S2-T1). But as I said earlier, it gets lost in the noise.... tan(0.1 degree) vs sin(0.1 degree)

So nothing more to hash out on the gun cant issue.

I already fixed the reference to tangent in the calculation. All the formulae in that spread sheet are yours. I will add the calculation of scope cant as a separate page so that one can solve those equations independently and then put the combined result into a graph and table on the third page. Might have that done this evening, or not ...

Thanks for all the help and patience in taking the time to work through that with me. Likes I said, I don't actually speak math, I just pretend to on the Internet.

Well done.

 

[Guest]

I think I spoke too quickly. Let's kick about scope cant a bit to make sure I understand what you are saying if you have time?

It seems to me scope cant, being centered on the POA and independent of drop from LOD, IS dependent upon zero range? That is to say effect of scope cant at zero range IS zero but at ranges above or below zero range there will be an error which is proportional to the cant angle and the height of the sight above the bore? I can calculate that because it is proportional to the dx,dy at the optic and the trajectory height / sight height above/below LOS. So it seems to me that shooters who dial (change their zero range) rather than use hold off would be mostly immune from scope cant so long as its presence did not cause gun cant. The likely hood of that would increase with range I would think. So using your drawing.

This drawing presumes you are shooting at your zero range and that the shooter does not unconsciously correct the scope cant thus causing gun cant in the other direction. Lets stipulate that the shooter is a machine rest and is not subconsciously correcting for scope cant.

Because the d(x), d IS a translation around the axis through POA on the target, if POA and POI coincide with the origin of the reticle, then there is no error. Under any other conditions we have hold off. That might be for wind or range but it's hold off which makes the problem apparent. Dialing the turrets changes the zero and so scope cant error CAN be dialed out. You would see evidence of the scope cant error when you calibrated your distance per click while setting up your optic. Horizontal and vertical clicks would not be equal. Given a large enough adjustment range you could actually measure reticle cant if you had everything else plumb by arctan(v-clicks/h-clicks) for a given distance.

How far have I wandered off from reality? I'll give it a rest now to see what you think.

 

[Scotchmo]

... So it seems to me that shooters who dial (change their zero range) rather than use hold off would be mostly immune from scope cant so long as its presence did not cause gun cant. The likely hood of that would increase with range I would think. So using your drawing.

... You would see evidence of the scope cant error when you calibrated your distance per click while setting up your optic. Horizontal and vertical clicks would not be equal. Given a large enough adjustment range you could actually measure reticle cant if you had everything else plumb by arctan(v-clicks/h-clicks) for a given distance.



How far have I wandered off from reality? I'll give it a rest now to see what you think.

That's the correct image. As you move up or down the canted reticle (or click the elevation turret), you are introducing a substantial horizontal error and a small elevation error.

"... So it seems to me that shooters who dial (change their zero range) rather than use hold off would be mostly immune from scope cant..."

Only true, if you have the opportunity to click the windage turret as well. So no advantage of clicking vs holdover when it comes to scope cant errors. Not a problem in benchrest where you can take sight-in shots for the one target distance you will shoot at. In FT, no sighter shots at the target and every shot is scored, so scope cant can be a problem.

"arctan(v-clicks/h-clicks) "

Normally, I do the longer (though less mentally intensive) route of narrowing in on the correct setting in small increments.

I have used that very technique with holdover to correct a canted scope in one go. Shoot and zero at 25 yards. Shoot a 10 yard shot and measure the horizontal and vertical POI via the reticle. If the horizontal is zero, scope rotation is good. If not, I use that arctan equation to calculate the angle, mark it on the scope tube, and rotate it that amount in the rings. - Done.

I always correct a canted scope. But if you want to know how much error an uncorrected scope would cause - the formulas should look very similar to the gun cant formulas except that drop amount is replaced by the compensation amount. I think they will look like this:

horizontal error = compensation x sin(scope_cant_angle)
vertical error = compensation x (1 - cos (scope_cant_angle))

The more elevation compensation required or the greater the scope cant angle, the greater the errors. A high scope requires more holdover or more click compensation at close range. A low scope requires more compensation at long range.